Some Arithmetic, Algebraic and Combinatorial Aspects of Plane Binary Trees

94 Pages·2014·3.18 MB·English

Some Arithmetic, Algebraic and
Combinatorial Aspects of Plane Binary Trees
Raazesh Sainudiin†
PartIwith:JenniferHarlow†&WarwickTucker(Maths@Uppsala-SW)
PartIIwith:SeanCleary(Maths@CCNY-USA),RobertGriffiths(Stats@Oxford-UK),
MareikeFischer(Maths&CompSci@Greifswald-DL)&DavidWelch(CompSci@Auckland-NZ)
†SchoolofMathematicsandStatistics,UniversityofCanterbury,
Christchurch,NewZealand
(OnSabbatical,Dept.ofMathematics,CornellUniversity,Ithaca,NewYork,USA
October 27 2014,
CornellDiscreteGeometry&CombinatoricsSeminar
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Part I: Arithmetic and Algebra over Plane Binary Trees
Main Idea & Motivating Examples
Regular Pavings (RPs)
Mapped Regular Pavings (MRPs)
Real Mapped Regular Pavings (R-MRPs)
Applications of Mapped Regular Pavings (MRPs)
Conclusions of Part I
Part II: Combinatorics for Distributions over Plane Binary Trees
Catalan Coefficients
Split-Path Invariant Distributions
Conclusions of Part II
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Part I: Arithmetic and Algebra over Plane Binary Trees
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2. naturally extends to
arithmetic over intervals, eg. [1,2]+[3,4] = [4,6]
3. Our Main Idea:
– is to further naturally extend to
arithmetic over mapped partitions of an interval called
Mapped Regular Pavings (MRPs)
4. – by exploiting the algebraic structure of partitions formed
by rooted-plane-binary (rpb) trees
5. – thereby provide algorithms for several algebras and their
inclusions over rpb tree partitions
Extending Arithmetic:
reals→intervals→mappedpartitionsofinterval
1. arithmetic over reals, eg. 1+3 = 4
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3. Our Main Idea:
– is to further naturally extend to
arithmetic over mapped partitions of an interval called
Mapped Regular Pavings (MRPs)
4. – by exploiting the algebraic structure of partitions formed
by rooted-plane-binary (rpb) trees
5. – thereby provide algorithms for several algebras and their
inclusions over rpb tree partitions
Extending Arithmetic:
reals→intervals→mappedpartitionsofinterval
1. arithmetic over reals, eg. 1+3 = 4
2. naturally extends to
arithmetic over intervals, eg. [1,2]+[3,4] = [4,6]
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4. – by exploiting the algebraic structure of partitions formed
by rooted-plane-binary (rpb) trees
5. – thereby provide algorithms for several algebras and their
inclusions over rpb tree partitions
Extending Arithmetic:
reals→intervals→mappedpartitionsofinterval
1. arithmetic over reals, eg. 1+3 = 4
2. naturally extends to
arithmetic over intervals, eg. [1,2]+[3,4] = [4,6]
3. Our Main Idea:
– is to further naturally extend to
arithmetic over mapped partitions of an interval called
Mapped Regular Pavings (MRPs)
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5. – thereby provide algorithms for several algebras and their
inclusions over rpb tree partitions
Extending Arithmetic:
reals→intervals→mappedpartitionsofinterval
1. arithmetic over reals, eg. 1+3 = 4
2. naturally extends to
arithmetic over intervals, eg. [1,2]+[3,4] = [4,6]
3. Our Main Idea:
– is to further naturally extend to
arithmetic over mapped partitions of an interval called
Mapped Regular Pavings (MRPs)
4. – by exploiting the algebraic structure of partitions formed
by rooted-plane-binary (rpb) trees
7/94
Extending Arithmetic:
reals→intervals→mappedpartitionsofinterval
1. arithmetic over reals, eg. 1+3 = 4
2. naturally extends to
arithmetic over intervals, eg. [1,2]+[3,4] = [4,6]
3. Our Main Idea:
– is to further naturally extend to
arithmetic over mapped partitions of an interval called
Mapped Regular Pavings (MRPs)
4. – by exploiting the algebraic structure of partitions formed
by rooted-plane-binary (rpb) trees
5. – thereby provide algorithms for several algebras and their
inclusions over rpb tree partitions
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arithmetic from intervals to their rpb-tree partitions
Figure: Arithmeticwithcolouredspaces.
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arithmetic from intervals to their rpb-tree partitions
Figure: Intersectionoftwohollowspheres.
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